energies
Article New Adaptive High Starting Torque Scalar Control Scheme for Induction Motors Based on Passivity
Juan Carlos Travieso-Torres 1,* , Miriam Vilaragut-Llanes 2 , Ángel Costa-Montiel 2, Manuel A. Duarte-Mermoud 3 , Norelys Aguila-Camacho 4 , Camilo Contreras-Jara 1 and Alejandro Álvarez-Gracia 1
1 Department of Industrial Technologies, University of Santiago de Chile, Santiago 9170125, Chile; [email protected] (C.C.-J.); [email protected] (A.Á.-G.) 2 Centro de Investigaciones y Pruebas Electroenergéticas (CIPEL), Universidad Tecnológica de la Habana “José Antonio Echeverría” (CUJAE), Havana 11500, Cuba; [email protected] (M.V.-L.); [email protected] (Á.C.-M.) 3 Departamento de Electricidad, University of Chile, Santiago 8370451, Chile; [email protected] 4 Departamento de Electricidad, Universidad Tecnológica Metropolitana, Santiago 7800002, Chile; [email protected] * Correspondence: [email protected]
Received: 14 December 2019; Accepted: 19 January 2020; Published: 10 March 2020
Abstract: A novel adaptive high starting torque (HST) scalar control scheme (SCS) for induction motors (IM) is proposed in this paper. It uses a new adaptive-passivity-based controller (APBC) proposed herein for a class of nonlinear systems, with linear explicit parametric dependence and linear stable internal dynamics, which encompasses the IM dynamical model. The main advantage of the HST-SCS includes the ability to move loads with starting-torque over the nominal torque with a simple and cost-effective implementation without needing a rotor speed sensor, variable observers, or parameter estimators. The proposed APBC is based on a direct control scheme using a normalized fixed gain (FG) to fine-tune the adaptive controller parameters. The basic SCS for induction motors (IM) and the HST-SCS were applied to an IM of 200 HP and tested using a real-time simulator controller OPAL-RT showing the achievement of the proposal goal.
Keywords: adaptive-passivity-based controller (APBC); scalar control scheme (SCS); induction motor (IM); proportional (P) controller; nonlinear dynamical systems
1. Introduction Compared to direct current electric motors, induction motors (IMs) have a lower cost and higher efficiency, require lower maintenance, and have been replacing them in variable speed operations with increasing use in the past twenty years [1]. In variable speed applications, the IM is fed by different types of alternating current (AC) drives [2], which differ in their performance with respect to the starting torque, transient speed behavior, and steady-state speed-accuracy of the desired speed [1]. For low, medium, and high-performance applications, three main control schemes are used: scalar control scheme (SCS) [3], direct torque control (DTC) [4], and field-oriented control (FOC) [5,6] respectively. There is also a simpler AC/AC soft starter based on thyristors [3], but it does not allow variable speed operation. It only reduces the starting stator current while moving low starting torque loads. This reduction is made by reducing the starting voltage applied to the motor. The AC drive with a rotor speed closed loop (CL), based on FOC or DTC, is used for high-performance applications, such as winding and takeoffs. Its capabilities consist of starting with torque around 150% of the nominal torque, having a rapid and no oscillatory transient speed
Energies 2020, 13, 1276; doi:10.3390/en13051276 www.mdpi.com/journal/energies Energies 2020, 13, 1276 2 of 15 behavior, and exhibiting 0.01% of steady-state speed accuracy. It needs parameter estimators to compute its characteristic variable slip frequency of operation. The AC drive based on sensorless DTC or sensorless FOC, is used for medium-performance applications. These consist of cranes, positive displacement pumps, and compressors. Its capabilities consist of starting with torque around 120% of the nominal torque, having a rapid and no oscillatory transient speed behavior, and 0.1% of steady-state speed accuracy. It needs parameter estimators and variable observers to compute its characteristic direct torque and flux control. In contrast, the cheapest AC-drive-based SCS is used for low-performance applications. Examples of these include blowers, fans, and centrifugal pumps. SCS only needs the rated voltage per phase and the frequency taken from the motor data plate. It moves loads needing a starting torque lower than 40% of the nominal torque [7] with 1% to 3% of steady-state speed accuracy and nonrapid and even oscillatory transient behavior [8–10]. These facts about SCS remain even if a CL speed control is considered as mainly improving steady-state accuracy [9]. This is the reason why FOC is finally proposed in [10] to reach a higher performance. Oscillatory transient behavior issues of the SCS are studied in detail in [11]. Several research works aim to improve the simplest SCS [12–17]. Compared to the DTC [4] and the FOC [5,6], the SCS needs neither parameter estimators nor estimator or sensor of the rotor speed. SCS schemes [3,12–17] assume that the electrical angular speed is equal to the reference rotor angular speed, neglecting the slip, or equals to the reference rotor angular speed plus an estimated slip. These are the simplest and most effective ways the estimated rotor angle is implemented. Improvements in works [12–15] consider estimating the slip to improve the rotor steady-state speed-accuracy. The electromagnetic torque is estimated in [12], and later the slip is based on this estimate. The gap power is estimated in [13] and then the slip is estimated, adjusting also the boost voltage. The slip is calculated based on a stator flux observer in [14], which depends on parameter estimators. A simpler slip calculation scheme is proposed in [15], where the computing depends on the rated stator current and the rated slip. The speed control of all these SCS schemes has an open-loop (OL) speed control. In addition, to improve the steady-state rotor speed accuracy of the SCS, a CL speed controller is proposed in [16]. The required electrical frequency is set by a CL speed fuzzy controller in [16]. It depends on the measurement of a rotor angular speed sensor. A rotor angular speed observer based on a spectral search method is proposed in [17] to improve the SCS speed monitoring. It is suggested in [17] that this monitoring could be used in other works to develop a CL sensorless SCS, but this is not fully addressed by [17]. All these schemes [12–17] aim to improve the accuracy of the steady-state rotor speed. They all need parameter estimators, depending on the accuracy of the estimates. The low starting torque issue of the SCS is not addressed in these references from [12–17]. It is in this paper where a solution is suggested. The mining and minerals industry, for instance, has machinery such as conveyor belts. These need high starting torque but no steady-state speed accuracy. One employed solution is to consider AC drives with a DTC or FOC scheme, paying an over cost. Another solution considers an AC drive with SCS to move the belt conveyors unloaded at the start. This last solution has an issue when unforeseen detentions occur. This paper aims to propose an alternative simple solution to move high starting-torque loads. It uses for current-control purposes the same current sensors used today by the basic SCS for motor protection. Compared to the DTC [4] and the FOC [5,6], the proposed high starting torque (HST)-SCS needs neither parameter estimators, variable observers, nor estimator or sensor of the rotor speed. As a result, it moves loads needing a starting torque around 100% of the nominal torque from 1% to 3% of steady-state speed accuracy and nonrapid and even oscillatory transient behavior [8–10]. It expands the applications of the SCS but is not capable of substituting the FOC nor DTC schemes. The main contribution of this paper is the proposal of an HST-SCS. It adds a CL adaptive-passivity-based controller (APBC) for current to the basic SCS. This is based on the adaptive theory from [18,19] developed for linear dynamical systems but extended herein for a class of nonlinear Energies 2020, 13, 1276 3 of 15 dynamical systems expressed in the normal form, which has a linear explicit parametric dependence with a linear internal dynamic and encompasses the IM model. The use of normalized FG is proposed to adjust the adaptive controller parameters. The paper is organized as follows: Section2 describes the IM model, the basic SCS method, and the APBC basis. The proposed APBC method for a class of nonlinear systems is proposed in Section3. Experimental results comparing basic SCS and the proposed HST-SCS are described in Section4. Finally, in Section5, some conclusions are drawn.
2. Preliminaries The design and performance of the proposed HST-SCS for IM will be compared to the basic SCS for the IM described in this section. Both schemes make use of several concepts and theories also described in what follows.
2.1. Basic SCS for IM The steady-state IM-equivalent circuit per phase is described in detail in Appendix A.1[ 20]. The basic SCS considers a P controller to keep constant magnetization at different electrical frequencies as shown in the following equation: √2p Vs_rated Vboost P1ωe∗ + √2Vboost, with P1 = for fmin < f < fc, 2π frated − f c Vs∗ = (1) √2p Vs_rated P2ω∗, with P2 = for fc < f < frated, e 2π frated where Vs∗ is the required amplitude of the phase stator voltage to be applied in order to achieve the required electrical angular frequency ωe∗. In practice, ωe∗ is considered to be equal to the required rotor angular speed multiplied by the pair of poles ωr∗p, neglecting the rotor angular slip. Vs_rated is the rated phase voltage from the motor data plate, p is the pair of poles, frated is the motor rated frequency in Hz, and the Vboost has a value of up to 25% of Vs_rated in [21] and operates from the minimum frequency fmin (with a value up to 6% of frated in [21]) to the low-frequency fc (with a value up to 40% of frated in [21]). The Vboost is a controller bias or offset needed to deliver a certain amount of starting torque according to Equation (A4): pV 2 Rrωslip T = 3 s (2) em 2 2 ωe 2 Rr + ωslipL0r
The scalar control aims to keep a constant flux modulus ( ϕ ), neglecting the stator voltage drop in the stator impedance (Rsis + jωeL is 0). After that, the Kirchhoff’s voltage law equation for the s0 ≈ stator (A1) was simplified and shows the dependence between the flux modulus ( ϕ ), the frequency ωe, and the modulus of the voltage applied to the motor, according to
Vs 0 ϕ z}|{ z }|≈ { z}|{ us us = (Rsis + jωeLs0is) + jωe Lmim => us jωeϕ => ϕ | | . (3) ≈ ∼ ωe
Thus, the amplitude of the rated voltage per phase and the rated electrical angular frequency are 2p √ Vs_rated related as P2 = 2π f rated = ϕrated , which is kept constant over fc throughout Equation (1). From this result, it can also be shown that at low speed, the required voltage Vs∗ = P2ωe∗ is low, and the stator impedance is no longer neglectable. This is the reason why a Vboost is applied under fc through the Equation (1) to assure the existence of flux and torque in this case. Finally, when operating at speeds higher than the rated speed ωe_rated, a saturator needs to be used after the P controller. It limits the stator voltage to the safe limit Vs_rated, thus protecting the IM. Energies 2020, 13, 1276 4 of 15
jρ After applying the Park transformation e− [22], the two-phase α–β instantaneous signals fixed at a stationary reference frame can be obtained from the d–q amplitude magnitudes rotating at the = jρ synchronous reference frame; thus, us∗ β e− Vsdq∗ . Similarly, from the α–β instantaneous signals, the ∝ jρ d–q amplitude magnitudes can be obtained through Isdq = e− is β. ∝ In addition, using the Clarke transformation T2 3 [23] from the two-phase α–β instantaneous → signals fixed at the stationary reference frame, the three-phasic instantaneous signals can be obtained Energies 2020, 13,h x FOR PEER REVIEWiT h i 4 of 14 through us = ua ub uc = T2 3u∗ and vice versa isαβ = isα isβ = T3 2is. → s β → TheThe generalgeneral diagramdiagram ofof thethe basicbasic ∝ SCSSCS forfor IMIM thatthat willwill bebe usedused inin thisthis paperpaper forfor comparisoncomparison purposespurposes isis shownshown inin FigureFigure1 1,, taken taken from from [ [21].21]. ItIt doesdoes notnot makemake useuse ofof parameterparameter estimatorsestimators whenwhen comparedcompared toto DTCDTC [[4]4] oror FOCFOC [[5,6],5,6], usingusing onlyonly thethe ratedrated voltagevoltage perper phasephase fromfrom thethe motormotor datadata plateplate andand sensorssensors of of the the stator stator current current consumption consumption to checkto check the the proper proper IM functioningIM functioning for itsfor protection. its protection. The ∗ setpointThe setpoint of the of required the required rotor angular rotor angular speed ω r∗speedis established 𝜔 is established by the operator, by the and operator, the SCS automaticallyand the SCS ∗ definesautomatically the required defines stator the required voltage V statorsd∗ and voltage electrical 𝑉 frequency and electricalωe. frequency 𝜔 .
FigureFigure 1.1. Simulink block diagram ofof basicbasic SCSSCS forfor thethe inductioninduction motormotor (IM).(IM).
TheThe SCSSCS schemescheme describeddescribed inin FigureFigure1 1 is is simple simple and and e effective.ffective. ItIt isis usedused forfor applicationsapplications needingneeding lowlow startingstarting torque torque and and a steady-statea steady-state speed speed accuracy accuracy up up to 3%.to 3%. To operatesTo operates motor motor loads loads needing needing HST, someHST, issuessome issues are faced are whichfaced which are described are described in the in next the section. next section. 2.2. Problem Statement 2.2. Problem Statement After applying the basic SCS for IM, the following issues are faced: After applying the basic SCS for IM, the following issues are faced: (a)(a) HST: To keepkeep thethe constantconstant fluxflux givengiven inin (3),(3), basicbasic SCSSCS usesuses thethe controllercontroller (1),(1), whichwhich hashas aa lowlow starting torque issue. This This is is due to the low appliedapplied startingstarting voltage,voltage, eveneven afterafter applyingapplying thethe Vboost, because the torque (2) is proportional to the square of the quotient between the phase rated 𝑉 , because the torque (2) is proportional to the square of the quotient between the phase ratedstator stator voltage voltage and rated and electrical rated electrical frequency. frequenc Thus,y. an Thus, alternative an alternative method controllingmethod controlling the required the stator current is∗ to have HST∗ is proposed next. required stator current 𝑖 to have HST is proposed next. (b) Nonlinear IM dynamical model with unknown parameters: To control the stator current i ∗, the (b) Nonlinear IM dynamical model with unknown parameters: To control the stator current 𝑖s∗ , the IM d-q dynamical model will be used (A5), whichwhich isis aa nonlinearnonlinear dynamicaldynamical systemsystem withwith linearlinear explicit parametric dependence and linear stable internal dynamics ((zz)) ofof thethe formform . y = A1 f1(y) + A2 f2(z) + Bu 𝑦 = 𝐴 𝑓 (𝑦) + 𝐴 𝑓 (𝑧) +𝐵𝑢 . (4) z = Cz + Dy (4) 𝑧 =𝐶𝑧+𝐷𝑦 where the flux, which is the internal dynamics variable 𝑧= 𝛹 𝛹 ∈ℜ , is not accessible, the current output variable 𝑦= 𝐼 𝐼 ∈ℜ and the voltage input (control) variable 𝑢= 𝑉 𝑉 ∈ℜ are both accessible. The system function 𝑓 (𝑦) = 𝑦 𝜔 𝑦 ∈ℜ is known, but 𝑓 (𝑧) = 𝑧 𝜔 𝑧 ∈ℜ the function is unknown. All system parameters are considered constant ´ 01 × but unknown, where A = − 𝐼 ∈ℜ , 𝐼 is the identity matrix of order 2, and A = −1 0 − 𝜔 01 × × × 𝐼 ∈ℜ , 𝐵 = 𝐼 ∈ℜ , 𝐶= ∈ℜ and 𝐷= 𝐼 ∈ −1 0 −𝜔 − ℜ × . B and C are invertible matrices. The sign(B) matrix contains the sign of each element of B and
Energies 2020, 13, 1276 5 of 15
T h i 2 where the flux, which is the internal dynamics variable z = Ψrd Ψrq , is not accessible, T ∈ < h i 2 the current output variable y = Isd Isq and the voltage input (control) variable T ∈ < T h i 2 h T T i 4 u = Vsd Vsq are both accessible. The system function f1(y) = y ωe y ∈ < T ∈ < h T p T i 4 is known, but the function f2(z) = z 2 ωrz is unknown. All system parameters ∈" < " ## Rs0 0 1 4 2 are considered constant but unknown, where A = I2 × , I2 is the 1 − σLs 1 0 ∈ < " " ## − RrLm Lm 0 1 4 2 1 2 2 identity matrix of order 2, and A2 = 2 I2 σL L × , B = σL I2 × , σLsLr s r 1 0 ∈ < s ∈ < − Rr ω Lr slip 2 2 RrLm 2 2 C = − R × and D = I2 × . B and C are invertible matrices. The ω r ∈ < Lr ∈ < − slip − Lr sign(B) matrix contains the sign of each element of B and is assumed to be known. Therefore, an adaptive controller designed for nonlinear systems of the form (4) is proposed herein. (c) Parameters adjustment of APBC: How to adjust the controller parameters K and the fixed-gain Γ [19] remains an open question today. We will try to answer it herein by considering a known system time constant and by including a normalized FG for the APBC.
3. HST-SCS Proposal As previously described, it can be seen from (3) that the flux modulus ϕ is proportional to the | | quotient of the phase rated stator voltage and the rated frequency (after neglecting the stator voltage drop in the stator impedance). However, from (3), it can also be seen that the flux modulus ϕ is | | proportional to the modulus of the magnetizing current im = is ir. Replacing ir from (A2), we get −
Rr + jLr0 jLm ωslip im = is is = is (5) Rr Rr − + jLr + jLr ωslip ωslip
Considering Equation (5), we explore how to keep a constant im by keeping a constant stator phase 2
= p jLm Rr 2 current is to guarantee an HST. This is obtained from (A3) and results as Tem 3 2 Rr ω is . +jLr slip ωslip | |
3.1. HST-SCS for IM A general diagram of the proposed HST-SCS is shown in Figure2. Here, a new HST control loop (marked in yellow) operating from the minimum frequency fmin to the lower frequency fc1 was added to the basic SCS. It moves the Vboost operation from fc1 to fc, keeping the P2ωe∗ operating from fc to frated. The new transition frequency fc1 from HST control to Vboost control is lower than fc. Energies 2020, 13, x FOR PEER REVIEW 5 of 14 is assumed to be known. Therefore, an adaptive controller designed for nonlinear systems of the form (4) is proposed herein. (c) Parameters adjustment of APBC: How to adjust the controller parameters K and the fixed-gain Γ [19] remains an open question today. We will try to answer it herein by considering a known system time constant and by including a normalized FG for the APBC.
3. HST-SCS Proposal As previously described, it can be seen from (3) that the flux modulus |φ| is proportional to the quotient of the phase rated stator voltage and the rated frequency (after neglecting the stator voltage drop in the stator impedance). However, from (3), it can also be seen that the flux modulus |φ| is proportional to the modulus of the magnetizing current 𝑖 =𝑖 −𝑖 . Replacing 𝑖 from (A2), we get